Physics of Heart: From macroscopic to microscopic Xianfeng Song Advisor: Sima Setayeshgar January 9, 2007

Outline  Part I: Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac: Insights From Physical Modeling  Part II: Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle  Part III: Calcium dynamics: exploring the stochastic effect

Part I: Transport Through the Myocardium of Pharmocokinetic Agents Placed in the Pericardial Sac: Insights From Physical Modeling Xianfeng Song, Department of Physics, Indiana University Keith L. March, IUPUI Medical School Sima Setayeshgar, Department of Physics, Indiana University

Diffusion in biological processes  Protein diffusion in single cells Elowitz, M. B., M. G. Surette, et al. (1999). "Protein Mobility in the Cytoplasm of Escherichia coli." Journal of Bacteriology 181(1):  Diffusion is important during early Drosophila embryonic pattern formation Gregor, T., W. Bialek, et al. (2005). "Diffusion and scaling during early embryonic pattern formation." Proceedings of the National Academy of Sciences 102(51):  Diffusion plays a crucial role in brain function Nicholson, C. (2001), “Diffusion and related transport mechanisms in brain tissue”, Rep. Prog. Phys. 64, science.howstuffworks.com web.jjay.cuny.edu

Pericardial Delivery: Motivation  The pericardial sac is a fluid-filled self-contained space surrounding the heart. As such, it can be potentially used therapeutically as a “drug reservoir.”  Delivery of anti-arrhythmic, gene therapeutic agents to  Coronary vasculature  Myocardium  Recent experimental feasibility of pericardial access  Verrier VL, et al., “Transatrial access to the normal pericardial space: a novel approach for diagnostic sampling, pericardiocentesis and therapeutic interventions,” Circulation (1998) 98:  Stoll HP, et al., “Pharmacokinetic and consistency of pericardial delivery directed to coronary arteries: direct comparison with endoluminal delivery,” Clin Cardiol (1999) 22(Suppl-I): I-10-I-16. V peri (human) =10ml – 50ml

Part 1: Outline  Experiments  Mathematical modeling  Comparison with data  Conclusions

Experiments  Experimental subjects: juvenile farm pigs  Radiotracer method to determine the spatial concentration profile from gamma radiation rate, using radio-iodinated test agents  Insulin-like Growth Factor ( 125 I-IGF, MW: 7734 Da)  Basic Fibroblast Growth Factor ( 125 I-bFGF, MW: Da)  Initial concentration delivered to the pericardial sac at t=0  200 or 2000  g in 10 ml of injectate  Harvesting at t=1h or 24h after delivery

Experimental Procedure  At t = T (1h or 24h), sac fluid is distilled: C P (T)  Tissue strips are submerged in liquid nitrogen to fix concentration.  Cylindrical transmyocardial specimens are sectioned into slices: C i T (x,T) x denotes C T (x,T) =  i C i T (x,T) x: depth in tissue i

Mathematical Modeling  Goals  Determine key physical processes, and extract governing parameters  Assess the efficacy of drug penetration in the myocardium using this mode of delivery  Key physical processes  Substrate transport across boundary layer between pericardial sac and myocardium:   Substrate diffusion in myocardium: D T  Substrate washout in myocardium (through the intramural vascular and lymphatic capillaries): k

Idealized Spherical Geometry Pericardial sac: R 2 – R 3 Myocardium: R 1 – R 2 Chamber: 0 – R 1 R 1 = 2.5cm R 2 = 3.5cm V peri = 10ml - 40ml

Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2005, Los Angeles Governing Equations and Boundary Conditions  Governing equation in myocardium: diffusion + washout C T : concentration of agent in tissue D T : effective diffusion constant in tissue k: washout rate  Pericardial sac as a drug reservoir (well-mixed and no washout): drug number conservation  Boundary condition: drug current at peri/epicardial boundary

Numerical Fits to Experiments Drug Concentration Conce Error surface

Fit Results Numerical values for D T, k,  consistent for IGF, bFGF

Time Course from Simulation Parameters: D T =7×10 -6 cm 2 s -1 k=5×10 -4 s -1  =3.2×10 -6 cm 2 s 2

Effective Diffusion,D *, in Tortuous Media  Stokes-Einstein relation D: diffusion constant R: hydrodynamic radius : viscosity T: temperature  Diffusion in tortuous medium D * : effective diffusion constant D: diffusion constant in fluid : tortuosity For myocardium,  = (from M. Suenson, D.R. Richmond, J.B. Bassingthwaighte, “Diffusion of sucrose, sodium, and water in ventricular myocardium, American Joural of Physiology,” 227(5), 1974 )  Numerical estimates for diffusion constants  IGF : D ~ 4 x cm 2 s -1  bFGF: D ~ 3 x cm 2 s -1  Our fitted values are in order of cm 2 sec -1, 10 to 50 times larger !!

Xianfeng Song, Indiana University, Bloomington, March APS Meeting 2005, Los Angeles Transport via Intramural Vasculature Drug permeates into vasculature from extracellular space at high concentration and permeates out of the vasculature into the extracellular space at low concentration, thereby increasing the effective diffusion constant in the tissue. Epi Endo

Diffusion in Active Viscoelastic Media Heart tissue is a porous medium consisting of extracellular space and muscle fibers. The extracellular space consists of an incompressible fluid (mostly water) and collagen. Expansion and contraction of the fiber bundles and sheets leads to changes in pore size at the tissue level and therefore mixing of the extracellular volume. This effective "stirring" results in larger diffusion constants.

Part I: Conclusion  Model accounting for effective diffusion and washout is consistent with experiments despite its simplicity.  Quantitative determination of numerical values for physical parameters  Effective diffusion constant IGF: D T = (1.7±1.5) x cm 2 s -1, bFGF: D T = (2.4±2.9) x cm 2 s -1  Washout rate IGF: k = (1.4±0.8) x s -1, bFGF: k = (2.1±2.2) x s -1  Peri-epicardial boundary permeability  IGF:  = (4.6±3.2) x cm s -1, bFGF:  =(11.9±10.1) x cm s -1  Enhanced effective diffusion, allowing for improved transport  Feasibility of computational studies of amount and time course of pericardial drug delivery to cardiac tissue, using experimentally derived values for physical parameters.

Part II: Electrical Wave Propagation in a Minimally Realistic Fiber Architecture Model of the Left Ventricle Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University

Part II: Outline  Motivation  Model Construction  Numerical Results  Conclusions and Future Work

Motivation  Ventricular fibrillation (VF) is the main cause of sudden cardiac death in industrialized nations, accounting for 1 out of 10 deaths.  Strong experimental evidence suggests that self- sustained waves of electrical wave activity in cardiac tissue are related to fatal arrhythmias.  Mechanisms that generate and sustain VF are poorly understood.  Conjectured mechanism: Breakdown of a single spiral (scroll) wave into a disordered state, resulting from various mechanisms of spiral wave instability. W.F. Witkowksi, et al., Nature 392, 78 (1998) Patch size: 5 cm x 5 cm Time spacing: 5 msec

Focus of this work Distinguish the role in the generation of electrical wave instabilities of the “passive” properties of cardiac tissue as a conducting medium  geometrical factors (aspect ratio and curvature)  rotating anisotropy (rotation of mean fiber direction through heart wall)  bidomain description (intra- and extra-cellular spaces treated separately) from its “active” properties, determined by cardiac cell electrophysiology.

From idealized to fully realistic geometrical modeling Rectangular slabAnatomical canine ventricular model Minimally realistic model of LV for studying electrical wave propagation in three dimensional anisotropic myocardium that adequately addresses the role of geometry and fiber architecture and is:  Simpler and computationally more tractable than fully realistic models  Easily parallelizable and with good scalability  More feasible for incorporating realistic electrophysiology, electromechanical coupling, J.P. Keener, et al., in Cardiac Electrophysiology, eds. D. P. Zipes et al. (1995) Courtesy of A. V. Panfilov, in Physics Today, Part 1, August 1996 bidomain description

Peskin asymptotic model: Fundamental principles and Assumptions  The fiber structure of the left ventricle is in near- equilibrium with the pressure gradient in the wall:  The state of stress in the ventricular wall is the sum of a hydrostatic pressure and a fiber stress  The cross-sectional area of a fiber tube does not vary along its length  The fiber structure has axial symmetry  The thickness of the fiber structure is considerably smaller than its other dimensions.

Peskin Asymptotic model: Conclusions  The fibers run on a nested family of toroidal surfaces which are centered on a degenerate torus which is a circular fiber in the equatorial plane of the ventricle.  The fiber are approximate geodesics on these surfaces, and the fiber tension is approximately constant on each surface  The fiber-angle distribution through the thickness of the wall follows an inverse-sine relationship.  The wall thickness is proportional to the radial distance of the middle surface from the symmetry axis.

LV Fiber Architecture  Early dissection results revealed nested ventricular fiber surfaces, with fibers given approximately by geodesics on these surfaces. Fibers on a nested pair of surfaces in the LV, from C. E. Thomas, Am. J. Anatomy (1957). Fiber angle profile through LV thickness: Comparison of Peskin asymptotic model and dissection results, from C. S. Peskin, Comm. in Pure and Appl. Math. (1989).  Peskin asymptotic model: first principles derivation of toroidal fiber surfaces and fiber trajectories as approximate geodesics

Model Construction  Nested cone geometry and fiber surfaces  Fiber paths  Geodesics on fiber surfaces  Circumferential at midwall subject to: Fiber trajectory: Fiber trajectories on nested pair of conical surfaces: inner surfaceouter surface

Governing Equations  Transmembrane potential propagation  Transmembrane current, I m, described by simplified FitzHugh-Nagumo type dynamics* v: gate variable Parameters: a=0.1,  1 =0.07,  2 =0.3, k=8,  =0.01, C m =1 * R. R. Aliev and A. V. Panfilov, Chaos Solitons Fractals 7, 293 (1996) C m : capacitance per unit area of membrane D: diffusion tensor u: transmembrane potential I m : transmembrane current

Numerical Implementation  Working in spherical coordinates, with the boundaries of the computational domain described by two nested cones, is equivalent to computing in a box.  Standard centered finite difference scheme is used to treat the spatial derivatives, along with first-order explicit Euler time-stepping.

Diffusion Tensor Local CoordinateLab Coordinate Transformation matrix R

Parallelization  The communication can be minimized when parallelized along azimuthal direction.  Computational results show the model has a very good scalability. CPUsSpeed up ± ± ± ± ± 0.85

Phase Singularities Color denotes the transmembrane potential. Movie shows the spread of excitation for 0 < t < 30, characterized by a single filament. Tips and filaments are phase singularities that act as organizing centers for spiral (2D) and scroll (3D) dynamics, respectively, offering a way to quantify and simplify the full spatiotemporal dynamics.

Filament-finding Algorithm Find all tips “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Random choose a tip “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Search for the closest tip “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Make connection “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue doing search “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm The closest tip is too far “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Reverse the search direction “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Continue “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Complete the filament “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Start a new filament “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding Algorithm Repeat until all tips are consumed “Distance” between two tips: If two tips are not on a same fiber surface or on adjacent surfaces, the distance is defined to be infinity. Otherwise, the distance is the distance along the fiber surface

Filament-finding result FHN Model: t = 2 t = 999

Numerical Convergence Filament Number and Filament Length versus Heart size  The results for filament length agree to within error bars for three different mesh sizes.  The results for filament number agree to within error bars for dr=0.7 and dr=0.5. The result for dr=1.1 is slightly off, which could be due to the filament finding algorithm.  The computation time for dr=0.7 for one wave period in a normal heart size is less than 1 hour of CPU time using FHN-like electrophysiological model. Fully realistic model requires several days per heart cycle on a high-performance machine (Hunter, P. J., A. J. Pullan, et al. (2003). "MODELING TOTAL HEART FUNCTION." Annual Review of Biomedical Engineering 5(1): )

Scaling of Ventricular Turbulence Both filament length These results are in agreement with those obtained with the fully realistic canine anatomical model, using the same electrophysiology. A. V. Panfilov, Phys. Rev. E 59, R6251 (1999) Log(total filament length) and Log(filament number) versus Log(heart size) The average filament length, normalized by average heart thickness, versus heart size

Conclusions and Future Work  We have constructed and implemented a minimally realistic fiber architecture model of the left ventricle for studying electrical wave propagation in the three dimensional myocardium.  Our model adequately addresses the geometry and fiber architecture of the LV, as indicated by the agreement of filament dynamics with that from fully realistic geometrical models.  Our model is computationally more tractable, allowing reliable numerical studies. It is easily parallelizable and has good scalability.  As such, it is more feasible for incorporating  Realistic electrophysiology  Biodomain description of tissue  Electromechanical coupling

Part III: Calcium Dynamics: Exploring the stochastic effect Xianfeng Song, Department of Physics, Indiana University Sima Setayeshgar, Department of Physics, Indiana University

Part III: Outline  Introduction to calcium dynamics in myocyte  Motivation: Why stochastic  Future work

Overview of Calcium Signals  Calcium serves as an important signaling messenger.  Extracellular sensing  Ca 2+ signaling during embryogenesis  The regulation of cardiac contractility by Ca 2+  Calcium sparks and waves Spiral Ca 2+ wave in the Xenopus oocyte. The image size is 420x420 um. The spiral has a wavelength of about 150 um and a period of about 8 seconds. Part B is simulation. Ca sparks in an isolated mouse ventricular myocyte. Mechanically stimulated intercellular wave in airway epithelial cells Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Fundamental elements of Ca 2+ signaling machinery  Calcium Stores: external stores and internal stores: Endoplasmic Reticulum (ER), Sarcoplasmic Reticulum (SR), Mitochondria  Calcium pumps: Ca 2+ is moved to these stores by a Ca 2+ /Na + exchanger, plasma membrane Ca 2+ pumps and SERCA pumps.  Calcium channels: Ca 2+ can enter the cytoplasm via receptor-operated channels (ROC), store- operated channels (SOC), voltage-operated channels (VOC), ryanodine receptors (RyR) and inositol trisphosphate receptors (IP 3 R). Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Ventricular Myocyte  The typical cardiac myocyte is a cylindrical cell approximately 100 um in length by 10um in diameter and is surrounded by a cell membrane known as the sarcolemma (SL)  Three physical compartments: the cytoplasm, the sarcoplasmic reticulum (SR) and the mitochondria.  The primary function of SR is to store Ca for release upon cellular excitation.  The junctional cleft is a very narrow space between the SL and the SR membrane.  The SR release channel, or ryanodine receptor (RyR) is found almost entirely within the part of the SR membrane which communicates with the juntional cleft. Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Ventricular Myocyte and Excitation-Contraction coupling  Ca-Induced Ca Release (CICR) 1.From the resting state (channel closed), Ca may bind rapidly to a relatively low affinity site (1), therby activating the RyR. 2.Ca may then bind more slowly to a second higher affinity site (2) moving the release channel to an inavitive state. 3.As cytoplasmic [Ca] decreases, Ca would be expected to dissocaiate from the lower affinity activating site first and then more slowly from the inactivating site to return the channel to the resting state.  Excitation-Contraction Coupling (ECC) A small amount of Ca is initiated by depolarization of the membrane, thus induce CICR, initiate contraction. Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Motivation: Why stochastic  The global Calcium wave are comprised by local release events, called puffs.  Binding kinetics is by itself a stochastic process.  Receptor number is small, i.e., Calcium sparks are thought to consist of Ca 2+ release from between 6 and 20 RyRs. (Rice, J. J., M. S. Jafri, et al. (1999). "Modeling Gain and Gradedness of Ca2+ Release in the Functional Unit of the Cardiac Diadic Space." Biophys. J. 77(4): )  Diffusive noise is large. The noise is limited by l is the effective size of receptors or receptor array. (W. Bialek, and S. Setayeshgar, PNAS 102, 10040(2005)) Schematic representation of a cluster of m receptors of size b, distributed uniformly on a ring of size a. From single localized Calcium response to a global calcium wave W. Bialek, and S. Setayeshgar, PNAS 102,10040(2005) Borisyuk, A. (2005). Tutorials in mathematical biosciences. Berlin, Springer

Future work

Thanks!!